September 22, 2003
Arlo Weil (Geology)
"What We're Counting, and Why"
Summary
Prepared by Anne Dalke
Additions, revisions, extensions are encouraged in the Forum
Participants
Arlo's "Random Walk of Counting Ancedotes" (PDF, 4.5 MB) began with a description of his own work in paleomagnetism (measurements of the earth's magnetic field which enable him to resconstruct where the continents were in the past) and the study of curved mountain belts. He observed that all measurement involves a "fudge factor," an uncertainty, and invited us to consider together the limits of this tool which we seem to presume is more "secure" than it really is. We keep trying to narrow the range, the "disagreement between measurement and 'true' or accepted value," but how do we know what is "true"? Terms like "accuracy" and "precision" are used interchangably, but have different meanings; producing additional samples ("more points") with the same distribution on a graph can appear (but not actually be) "more meaningful." There is a difference between "error" (the disagreement between measurement and "true" or accepted value) and "uncertainty" (the interval around a measured value).
Arlo then offered a range of examples of "how we all count wrong." For instance, we use two different number systems. Most common is the form of "tallying" which uses 1-10 as cardinal elements (and comes naturally to children learning to count on their fingers). An alternative is the base-10 system, which includes zero (a late addition in the history of mathematics--"why are we afraid of zero?"). Arlo also suggested that we re-consider how many hours are in a day: We think it lasts 24 hours, but it "actually" takes 48 hours for the world to "fill up" and then to "empty" of a "Tuesday." Because of our various methods of counting, we were confused about, too, which year the millenium turned; similar confusions attended the shift from the Julian to the Gregorian calendar. Such differences are intrinsic to the fact that we count in a variety of ways; the answer to the query, "what is the difference between 15 and 21?" could be 7 (if one "counts up") or 6 (if one subtracts); the answer seems to be an index, in the first place, to the intervals between the two numbers, and in the second to the tally of numbers between them.
Arlo also suggested the importance of taking note of the different contexts, and possible political motivations, for the different ways in which we count. The procedures involved in "counting the huddled masses" include a wide range of methods for estimating or producing "best guesses." What is the best way of counting the number of moose in a certain area, or how many people have gathered for a demonstration? (By taking a cross section of the area, estimating the flux, and then multiplying? By taking an arial photo and then extrapolating from the numbers observed?) How best answer the question, "If the land surface of earth was divided evenly, how much could each of us call our own?" Debates about how to get the most accurate census figures were also rehearsed: how statistically useful is representative sampling? (The long-standing political practice of gerry-mandering was also described.) What is the "truth value" of longitudinal studies about race? These numbers, which can vary in orders of magnitude, have very important policy implications.
Observing that monkeys have been said to have a "language-independent ability to discriminate numbers," Arlo then turned to the question of how different forms of counting have countributed to paradigm shifts. How we have come to understand evolution (in light of the gaps in the fossil record)? How we have come to understand the existence of brown dwarfs and missing cosmic masses? How have different ways of counting affected our understanding of the frequency of disasters (as more data has been gathered and reported by the Center for Research on the Epidemiology of Disaster and the Office of Foreign Disaster Assistance)? How does our understanding of different eras of history vary, as our measurements shift from geologic to cellular scales of time? Arlo ended his presentation with an evocation of "counting the unimaginable": a reminder of the scale needed to describe the evolution of a star and "how far" its beginning is from "where we are today."
During discussion, conversation returned to the question of whether one scale is "more real" than another, to the observation that what might matter most is "locality" or "proximity" ("how much do we care about ourselves as the standard from which all measurements are made?"), and to our inclination to "weighted averages." Do we actually operate unconsciously on a "rhythmic" or log scale, which comes to "seem unnatural" as we train ourselves (and our children and students) to make counting a conscious activity?
A discussion of "Conforming to the Count: How We Decide to Do the Things We Do," will continue at noon next Monday, September 29, when Elliot Shore (from Information Services) will lead a conversation about decision-making in which we will share our own experiences of conforming to the count, of times when we have resisted doing so--and why.
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